Sturmey tips from Mark Stonich.

Lubrication:

Modern factory Sturmey greases: gear, SA103A
bearing, SA103B

Sturmey SA103A is a semi-fluid, NLGI #00 grease, a product that has been developed for industry to replace oils in gearboxes either for longer service life or because the gearbox has developed a leak. That should do fine, if only you can find a small quantity.

How about oil? I've seen the following recomended, all of which seem reasonable:
20wt motor oil
30wt motor oil
10W-30wt synthetic motor oil
automatic transmission fluid
75W-90 gear oil
75W-90 synthetic gear oil
sewing machine oil

Of these I'm personally attracted to the 75W-90 synthetic gear oil. It's locally available in quarts, it's not terribly expensive, it has extremely long shelf and service lives, it's designed for use in non-pressurized lubrication systems and as a purpose designed gear lubricant it has anti-sheer additives.

HTH,
tcs